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Lecture Notes in Management Science (2014) Vol. 6: 207–216 6th International Conference on Applied Operational Research, Proceedings © Tadbir Operational Research Group Ltd. All rights reserved. www.tadbir.ca

ISSN 2008-0050 (Print), ISSN 1927-0097 (Online)

A fast heuristic for the prize-collecting

Steiner tree problem

Murodzhon Akhmedov, Ivo Kwee and Roberto Montemanni

Dalle Molle Institute for Artificial Intelligence IDSIA-USI/SUPSI Galleria 2, CH-6928 Manno, Switzerland {murodzhon, roberto}@idsia.ch; [email protected]

Abstract. The Prize-Collecting Steiner Tree Problem (PCSTP) is a generalized version of the Steiner Tree Problem. PCSTP is well known and well studied problem in Combinatorial Optimization. Since PCSTP is NP-hard, it is computationally costly to achieve solutions for large instances. However, many real life network problems come with a wide range of variables and large instance sizes. Therefore, there is a need for efficient and fast heuristic algorithms to discover the hidden

knowledge behind vast networks. There exists a fast heuristic algorithm for the Steiner Tree Problem in the literature, which is based on Minimum Spanning Trees. In this paper, we propose to extend the existing heuristic algorithm to solve PCSTP. The performance of the extended heuristic (MST-PCST) is evaluated on available benchmark instances from the literature. We also test MST-PCST on randomly generated huge graph instances with up to 40000 nodes and 120000 edges. We report the average gap percentage between the solutions of MST-PCST and existing solution approaches in the literature. Results show that overall performance of MST-PCST is promising with tolerable gap percentage and reasonable running time on larger instances. It has a significantly faster running

time when graphs scale up which can shed light on large real world network instances.

Keywords: prize-collecting steiner tree problem; minimum spanning tree; combinatorial optimization

Introduction

The Steiner Tree Problem (STP) is a popular and well-studied problem in Graph Theory

as well as in Combinatorial Optimization. So-called Prize-Collecting Steiner Tree Problem

(PCSTP) is a generalized version of STP. Given an undirected network G = (V, E) where

nodes and arcs of network are respectively associated with node prizes and arc

costs . For simplicity, the set of nodes with are called terminal nodes and

the rest of the nodes as Steiner nodes (Tuncbag et al, 2012). PCSTP finds a sub-graph

G′ = (V′, E′) of G where it minimizes the total arc costs in the sub-graph and prizes of

208 Lecture Notes in Management Science Vol. 6: ICAOR 2014, Proceedings

nodes that are not in the sub-graph. This problem corresponds to minimization of the

following equation and it is easy to see that every optimal sub-graph will have a tree

structure.

This is also known as the Geomans − Williamson Minimization problem (Johnson et

al, 2000) in the literature. PCST has a variety of applications in utility networks such as

making fiber optic, gas or district heating connections for households economically feasible.

Recently a connection between PCSTP and biological networks has been identified

(Becheta et al, 2010). Briefly, in a large gene-gene or protein-protein interaction, network

PCSTP attempts to find a neighborhood or sub-network where genetic aberrations are most concerned. For example using gene expression data, one can find a connected

neighborhood where many genes are differently expressed.

Since PCSTP is NP-hard it is time consuming or in some cases it is impossible to obtain

a solution for large instances in a reasonable time. However, real life problems from

biology, for instance, come with a huge number of variables and instance sizes. We aim

to perform functional analyses of genes by using gene expression profile networks and

gene interaction networks, and usually the size of these networks can be very large with

up to 20,000 nodes and more than 100,000 arcs. In order to attain our goal, these giant

networks should be analyzed in a reasonable time. The existing exact solution approaches

in the literature suffer from solving the problem when the input graph scales up. Thus,

we need efficient and fast heuristic algorithms to interpret the hidden knowledge behind

the giant biological networks. There exists a fast heuristic algorithm for STP in the literature, which is based on Minimum Spanning Tree (Kou et al, 1981). For STP, it is

proven that the heuristic has an approximation ratio of 2(1-1/l) where l is the number of

leaf nodes in the optimal tree. In this paper, we propose to extend the existing heuristic

algorithm to solve PCSTP and enrich the heuristic with an effective leaf node pruning

strategy (MST-PCST). However, it is not guaranteed yet that the heuristic still preserves

the approximation ratio or not. The remainder of the paper is organized as follows: Related

work in the literature is discussed in next section. Then, the extended MST-PCST heuristic

approach is described in section 3. The fourth section consists of computational results

followed by conclusions and future work.

Related Work

There have been developed exact solution approaches as well as heuristics to solve

PCSTP in the literature. The PCSTP was introduced by Bienstock et al (1993). Segev

(1987) proposed the node weighted Steiner tree problem where a predetermined set of

nodes should be included into the final tree solution. Exact methods were de et al (2005) and (2006) where PCSTP was formulated as a mixed integer linear

programming, and a branch-and-cut algorithm was proposed as a solution methodology.

The quota version of PCSTP was studied by Haouari et al (2010) in which the budget or

quota limit was considered within the constraint set. Canuto et al (2001) proposed a heuristic

M Akhmedov et al 209

local search with perturbations. Johnson et al (2000) investigated a different variant of

PCSTP and proposed a strong pruning rule for primal-dual 2-approximation algorithm. Klau

et al (2004) studied the combination of a memetic algorithm with integer programming.

Recently a robust optimization approach was devised for PCSTP by Miranda et al

(2013). Mixed integer linear programming models for similar problems, the minimum

power broadcasting and multicasting problems, have been proposed by Barta et al

(2010), and Montemanni et al (2008, 2011), and Montemanni and Leggieri (2010).

Methodology

In this section we demonstrate our MST-PCST heuristic algorithm to solve PSCTP for the

Goemans-Williamson minimization function. Given an undirected network G = (V, E),

the set of terminal nodes can be identified, which consists of nodes with . One is

asked to identify a sub-graph of G according to the minimization function in equation (1).

By analyzing the objective function it is easy to notice that not all terminal nodes necessarily

should be included into the output tree structure.

PCSTP is a generalized version of the Minimum Spanning Tree (MST) problem. In

contrast to PCSTP, all nodes are considered as terminal nodes in MST. There exist fast

and efficient algorithms to solve MST. The pseudocode of a proposed MST-PCST heuristic

is presented in Algorithm 1. MST-PCST targets to obtain a solution for PCSTP with

large network instances in a reasonable time. The intuition behind the algorithm is

likening the PCSTP to the MST problem and employing existent efficient algorithms to

solve huge instances. Simply, MST-PCST reduces the vast network to a smaller network


with particular nodes and arc costs, solves MST on a smaller artificial network, and projects

the solution obtained in the smaller network back to the large network. The heuristic

consists of two phases.

Each step of the algorithm is illustrated below along with its explanation. After the

initialization of variables, we focus on the first phase of the heuristic to analyze. As a first

t rat w th th ‘wh ’ , th a g r thm c str cts a c m t gra h G’ = (V’, E’)

from G where V’ consist of all terminal nodes. Please see Figure 1 for further clarification.

Each arc length between terminal nodes in E’ corresponds to a shortest path length between

those terminal nodes in the original network G (Fig. 1 B). The positions of the terminal

nodes are reorganized in the figure for clarity. The shortest path problem is solved by

D kstra’s a g r thm (Dijkstra, 1959). All nodes on the shortest path between terminal

nodes in graph G are recorded for sub-graph construction in later stages. Afterward the

heuristic solves MST on G’ y m y g Pr m’s a g r thm (Prim, 1957) (Fig. 1 C).

The total cost of arcs in the resulting MST tree is recorded as C. Then the tree solution

in G’ is converted into the original graph G (Fig. 1 D). It is possible that some arcs can

appear more than once in G after this conversion. In this case they are reduced to a single

arc. It is guaranteed that we again obtain a tree in G after this conversion since each arc

G’ corresponds to a path.

Fig. 1. MST-PCST Algorithm: Iteration 1. Yellow nodes correspond to terminal nodes, black nodes represent Steiner nodes, and arc widths represent the arc costs


Fig. 2. MST-PCST Algorithm: Iteration 2

Th s c t rat f th ‘wh ’ s m strat F g r 2. N w th r a ready

exists a sub-graph consisting of all terminal nodes and possibly some Steiner nodes in G,

which is obtained in the previous iteration (Fig. 2 D). In this iteration, the algorithm again

constructs a complete graph G’ in the same way as explained before where G’

= (V’, E’)

and V’ includes all nodes of the sub-graph in G (Fig. 2 E). Then MST is solved on G’

and the total cost of arcs in the resulting MST tree is recorded as C’ (Fig. 2 F). If C’

< C,

then the resulting tree is converted into the original network (Fig. 2 G). This process,

generating G’ from G, solving MST on G’

and converting the resulting tree back into G

is continued until the algorithm converges, meaning it is not possible to obtain a tree

with cost of C’ < C. No other Steiner node could be added to G’

anymore and the decrease

in C’ is monotonic, because at each iteration the algorithm is looking for a new spanning

tree with lesser cost that consists of nodes in a recent tree and probably some other

nodes. The logic behind this process is generating a tree with as low as possible total arc

cost while keeping the total node prize high by maintaining all terminal nodes in a tree.

Then, this phase is terminated and the algorithm continues with the pruning phase.

In the second phase of the heuristic each leaf node of sub-graph in G is tested for

pruning. If the prize of the leaf node is smaller than the connection cost, that node is

simply eliminated from the sub-graph. The connection cost could be a single arc cost or

cost of the path in the case where a terminal node is connected to a sub-graph via several

Steiner nodes. The final solution of the MST-PCST algorithm is presented in Figure 3 that is a pruned version of the graph in Fig. 2 G.


Fig. 3. Final sub-graph: A Tree

Computational Results

This section consists of computational studies on two sets of problem instances. The

performance of MST-PCST is evaluated on benchmark instances in the literature and

randomly generated large instances. We compare the running time of the MST-PCST

heuristic with the running times of existing approaches in the literature. Detailed information

about instances and running time data of approaches are reported later in this section.

The first problem set is the benchmark instances that are available in the literature.

The C and D instance series from Canuto et al (2001 , a g a g 2 r a

w r sta c s fr m et al (2005) are used for computational studies. The C and

D instances are generated from the Steiner problem of the OR-Library (Beasley, 1990) and detailed descriptions of generation can be found in Canuto et al (2001). The Cologne1

and Cologne2 real world instances have been used for designing the fiber optic networks of

a German city. These instances are generated based on real network infrastructure and

ta scr t s f g rat ca f et al (2005). The C set contains

40 instances with 500 nodes and 625-12500 edges, and D set contains 40 graphs with

1000 nodes and 1000-25000 edges. Cologne1 and Cologne2 consist of total 35 instances

with 768-1819 nodes and 69077-213973 edges. Canuto et al (2001) is a multi-start local

search algorithm for PCSTP where the initial solution is generated by a primal-dual

algorithm and a variable neighborhood search is utilized in a post-optimization proce r .

et al (2005) is an exact solution approach for PCSTP where the problem was

formulated as a MILP and the branch-and-cut algorithm construction was proposed to solve the MILP. In their implementation, they associated the connectivity inequalities

with the cuts in a directed graph, and found the violating inequalities by using a maximum

flow algorithm.

The second set of problem instances consists of 30 randomly generated huge graphs

with n nodes (ranging from 5625 to 40000) and m edges (ranging from 16875 to

120000). Since our goal is devising the heuristic algorithms to analyze the giant real

world instances, we would like to see the performance of MST-PCS when the graph

scales up. The generated graphs that consist of n nodes have m edges with corresponding

values of 3n and 6n, respectively. The instances are generated in the following manner.


Each integer point in a Cartesian plain in the range of [1, k] in the x-axis and in the

range of [1, k] in the y-axis represents a single node where k is an integer value. Basically,

k2 gives the number of nodes n in a graph. At the beginning, we build a random spanning

tree that contains all nodes to maintain the connectivity of the graph. Then we randomly

select two nodes and add an edge, if there is no edge between them. The edge between

node i and j with the corresponding coordinates of (a1, b1) and (a2, b2), respectively, is

added to the graph with the cost of cij and cost is determined as follows:

Associating the prize with the terminal nodes is nontrivial. Especially, prizing the

node is the issue in real world network examples from the biology. The prize of terminal

nodes should be proportional to the cost of edges in the graph in order to obtain a

meaningful output tree. The maximum possible edge cost in the generated graph could

be In our implementation, the terminal nodes are given

prizes from the discrete uniform distribution in We used

three different couples in graph generation and those are: (0.1, 0.5), (0.6, 0.8)

and (0.8, 1.0). The terminal nodes are randomly selected among the nodes and each

randomly generated graph is analyzed with 150 terminal nodes. The new instances are

available upon request to the authors. Since the computational studies of approaches are performed on different machines,

we use the scaling factor to bring the running time data of approaches into fairly equivalent

base in order to compare the performance. For comparing MST-PCST running time

w th th r s ts f et al (2005) (obtained on a Pentium IV with 2.8 GHz, 2 GB RAM,

SPECint2000 = 1204) and Canuto et al (2001) (achieved by Pentium II with 400 MHz,

64 MB RAM) the scaling factor is obtained from Dongarra (2013). The computational

studies for MST-PCST were performed on 2.9 GHz MacBook with 8 GB of memory.

However, Dongarra (2013) does not provide a direct scaling factor for these three machines.

We can establish a conservative estimate by taking the performance of similar machines

into account (Intel Pentium IV 2.8 GHz with 1317 Mflops/s) and (Intel Pentium II 333

MHz with f s s . D g th r g t m f et al (2005) by factor of 12 and Canuto et al (2001) by factor of 50 gives a reasonable basis of comparison to our

running time data. The MST-PCST algorithm is implemented in C++ environment and

Boost Graph Library (Siek et al, 2000) is employed.

Figure 4 demonstrates the scatter plot of the number of nodes in a graph versus the

running time of the various approaches. et al (2005) is an exact solution methodology

that is formulated as a MILP and provides the optimal solutions. For this approach, we

do not have figures for when the best solutions (not proven to optimality) are retrieved.

So, we report the time needed to obtain optimal solutions in the figure. Canuto et al

(2001) is a multi-start local search algorithm for PCSTP. From the figure, it is clear that

the r at tw th sta c s a th r g t m f et al (2005) and

Canuto et al (2001) s t a . c th et al (2005) solution was formulated

based on a MILP, the number of variables become huge when the graph scales up and it is computationally costly to use this approach in large instances. However, the running

time of MST-PCST is more preferable and this supports that the extended heuristic

could be useful to interpret the instances with a large number of nodes and edges.


Fig. 4. The number of nodes in input graph versus MST-PCST algorithm running time plot

The Table 1 summarizes the average gap percentage of results obtained by MST-PCSTP

with respect to the optimal solutions. The first and third rows of the table provide the

information about the size of test instances. The last two instance sets in the third row

belong to Cologne 1 and 2 test sets, and each corresponding gap value demonstrates the

average optimality gap of 15 instances. The rest of the instances refer to C and D sets,

and each corresponding gap value indicates the average optimality gap of 10 instances.

Although MST-PCST is a simple heuristic, it demonstrated a good performance on these

instances with the tolerable gap percentages. From the table it is easy to notice that MST-PCSTP achieved solutions with smaller optimality gap on a larger graph instances.

Furthermore, the heur st c’s fast r r g t m th arg r sta c s mak s t a an-

tageous to analyze the huge networks from real-world applications.

Table 1. The average optimality gap of MST-PCSTP on test instances C, D, and Cologne 1, 2

(V, E) (500, 625) (500, 1000) (500, 2500) (500, 12500) (1000, 1250)

Gap (%) 1.53 3.37 3.51 8.47 1.80

(V, E) (1000, 2000) (1000, 5000) (1000, 25000) (768, 69077) (1819, 213973)

Gap (%) 3.32 2.97 9.49 0.81 0.69

Conclusion

t a s m h r st c t s P TP th s st y. Th a r ach s t st chmark sta c s f et al (2005) and Canuto et al (2001) that are

available in the literature, and randomly generated instances. The results obtained by

MST-PCST demonstrated that the overall performance of the heuristic is good with a

slight deviation from optimum. MST-PCST has a fast running time on real world instances

and randomly generated large instances, therefore it has a potential to investigate the giant


real world instances. As future work, we plan to enrich the MST-PCST algorithm with

local search with some perturbations in order to improve the objective value. Furthermore, it

also can be integrated with Mixed Integer Linear Programming to achieve better results.

Acknowledgments— M. A. is supported by Swiss National Science Foundation through project 205321- 147138/1: "Steiner Trees for Functional Analysis in Cancer System Biology".

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